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Vietnam Journal of Mathematics 34:2(2006) 171-178

K₀ of Exchange Rings with Stable Range 1

Xinmin Lu and Hourong Qin

Abstract.  A ring R is called weakly generalized abelian (for short, WGA-ring) if for each idempotent e in R, there exist idempotents f,g,h in R such that $eR\cong fR\oplus gR$ and $(1-e)R\cong fR\oplus hR$, while gR and hR have no isomorphic nonzero summands. By an example we will show that the class of generalized abelian rings (for short, GA-rings) introduced in [10] is a proper subclass of the class of WGA-rings. We will prove that, for an exchange ring R with stable range 1, K₀(R) is an $\ell$-group if and only if R is a WGA-ring.

2000 Mathematics Subject Classification: 19A49, 16E20, 06F15.

Keywords: K₀-group, exchange ring, weakly generalized Abelian ring, Stable range 1, $\ell$-group.

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Published by Springer since January 2013